Numerical methods for mass transfer in falling films and two-phase flows with moving contact lines

The main goal of this thesis is to study numerical methods for the simulations of two classes of problems in fluid dynamics, namely mass transfer problems in falling films and two-phase immiscible flows with moving contact lines (MCL).A falling film is a thin layer of liquid flowing down an inclined or vertical wall. In many industrial applications, the liquid phase occurs as a gravity driven thin film. The direct numerical simulation of the fluid dynamics of the falling film based on two-phase models requires massive computational effort and sophisticated numerical tools, which limits its use in practice. Because of this, a new reduced one-phase transport model for liquid falling films is presented. An arbitrary Eulerian-Lagrange (ALE) approach is introduced to handle the free surface, which is moving in time in this reduced model. An isoparametric finite element method based on the ALE approach is used to approximate the curved surface accurately, and a so-called Streamline-Upwind Petrov-Galerkin stabilization technique is added to deal with the dominance of convection.A Navier boundary condition consists of a combination of a Dirichlet and a Robin boundary condition in different directions on the sliding wall. To enhance flexibility, the Dirichlet boundary condition is imposed weakly by Nitsche's method for Stokes flows with Navier boundary conditions, which is by itself an interesting topic and also serves as a preparation step for MCL flow problems. An analysis of Nitsche’s method in a polygonal domain based on the theory of saddle point problems is presented. For the problem in a curved domain, a Babuska-type paradox may occur. This paradox roughly means that the solution of a Stokes problem with a slip type boundary condition is not the limit of the solutions to the same equations posed on polygonal domains approaching the curved domain.An isoparametric finite element method and a method based on special treatment of the outer normals of curved boundary are presented to circumvent this paradox. For two-phase flows with MCL, we consider a (large) class of two-phase sharp interface models in which the interface and MCL modeling is based on constitutive laws for the interface stress tensor and for effective wall and contact line forces. A general variational formulation of this class of problems is derived, and an efficient level set based finite element discretization method is developed. In the variational formulation there are surface and contact line functionals, resulting from the natural interface and boundary conditions. We treat a general approach for an accurate discretization of these functionals based on a higher order finite element approximation of the level set function. A stabilized extended finite element method (XFEM) is used to handle the pressure discontinuity. By extending the method used in Stokes flows, the no-penetration boundary condition on the sliding wall is treated as a natural boundary condition using the Nitsche technique.The aforementioned numerical methods have been implemented in the software package DROPS, which is developed at the Chair for Numerical Mathematics at RWTH Aachen University. The accuracy of the methods is investigated in test cases with prescribed analytical solutions. For the mass transfer problems in falling films, simulated results are obtained in a realistic experimental setting and compared with measurement data. From the class of MCL flow problems, droplet sliding problems, Couette flows, wetting on a chemically patterned plate and a water droplet impact problem have been simulated to demonstrate the performance of the numerical solver